A210748 - OEIS

%I #5 Mar 30 2012 18:58:17

%S 1,3,2,6,10,5,11,29,32,13,19,71,118,99,34,32,156,352,437,299,89,53,

%T 322,919,1521,1526,887,233,87,636,2205,4559,6036,5117,2595,610,142,

%U 1218,4979,12373,20320,22591,16653,7508,1597,231,2279,10751,31233

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210747; see the Formula section.

%C Row n starts with -2+F(n+3) and ends with F(2n-1), where F=A000045 (Fibonacci numbers).

%C Row sums: A002450

%C Alternating row sums:  1,1,1,1,1,1,1,1,1,...(A000012)

%C For a discussion and guide to related arrays, see A208510.

%F u(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,

%F v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 3....2

%e 6....10...5

%e 11...29...32....13

%e 19...71...118...99...34

%e First three polynomials v(n,x): 1, 3 + 2x, 6 + 10x +5x^2

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

%t v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]   (* A210747 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]   (* A210748 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)

%Y Cf. A210747, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 25 2012